Painlev e monodromy varieties: geometry and quantisation Volodya - - PowerPoint PPT Presentation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e monodromy varieties: geometry and quantisation

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. arXiv:1511.03851 Talk at the RAQIS-2016 (Conference ”Recent Advances in Quaintum Integrable Systems”) University of Geneva, August, 26, 2016

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Plan:

Painlev´ e equations, Confluence, Isomonodromy and Affine cubics; Decorated character varieties; Quantisation;

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev` e list

d2w dz2 = 6w 2 + z d2w dz2 = 2w 3 + zw + α d2w dz2 = 1 w (dw dz )2 − 1 z dw dz + αw 2 + β z + γw 3 + δ w d2w dz2 = 1 2w (dw dz )2 + 3 2w 3 + 4zw 2 + 2(z2 − α)w + β w d2w dz2 = 3w − 1 2w(w − 1)w 2

z − 1

z dw dz + γw z + (w − 1)2 z2 αw 2 + β w + δw(w + 1) w − 1 d2w dz2 = 1 2 ( 1 w + 1 w − 1 + 1 w − z ) w 2

z −

(1 z + 1 z − 1 + 1 w − z ) wz + +w(w − 1)(w − z) z2(z − 1)2 [ α + β z w 2 + γ z − 1 (w − 1)2 + δ z(z − 1) (w − z)2 ]

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e transcendents - paradigmatic integrable systems

Reductions of soliton equations (KdV, KP, NLS);

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e transcendents - paradigmatic integrable systems

Reductions of soliton equations (KdV, KP, NLS); They admit a Hamiltonian formulation;

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e transcendents - paradigmatic integrable systems

Reductions of soliton equations (KdV, KP, NLS); They admit a Hamiltonian formulation; They can be expressed as the isomonodromic deformation of some linear differential equation with rational coefficients;

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e transcendents - paradigmatic integrable systems

Reductions of soliton equations (KdV, KP, NLS); They admit a Hamiltonian formulation; They can be expressed as the isomonodromic deformation of some linear differential equation with rational coefficients; Recently: PII - has a genuine fully NC analogue (V. Retakh-V.R.)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e transcendents - paradigmatic integrable systems

Reductions of soliton equations (KdV, KP, NLS); They admit a Hamiltonian formulation; They can be expressed as the isomonodromic deformation of some linear differential equation with rational coefficients; Recently: PII - has a genuine fully NC analogue (V. Retakh-V.R.) More recently: PIV - has a (non genuine) fully NC analogue (M. Cafasso-M. Iglesias)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Confluences of the Painlev´ e equations-1

PIII

• PD7

III

• PD8

III

PVI

PV

• Pdeg

V

• PJM

II

PI

PIV

• PFN

II

• Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´

e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Confluences of the Painlev´ e equations-2

Example Take w(z) = ϵ ˜ w(˜ z) + 1

ϵ5 ,

z = ϵ2˜ z −

6 ϵ10 ,

α =

4 ϵ15 then PII

d2w dz2 = 2w3 + zw + α becomes d2 ˜ w d˜ z2 = 6 ˜ w2 + ˜ z + ϵ2(2 ˜ w3 + ˜ z ˜ w), that for ϵ → 0 is PI.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

All Painlev´ e equations are isomonodromic deformation equations (Jimbo-Miwa1980) dB dλ − dA dz = [A, B] A = A(λ; z, w, wz), B = B(λ; z, w, wz) ∈ sl2.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

All Painlev´ e equations are isomonodromic deformation equations (Jimbo-Miwa1980) dB dλ − dA dz = [A, B] A = A(λ; z, w, wz), B = B(λ; z, w, wz) ∈ sl2. This means that the monodromy data of the linear system dY dλ = A(λ; z, w, wz)Y are locally constant along solutions of the Painlev´ e equation.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

All Painlev´ e equations are isomonodromic deformation equations (Jimbo-Miwa1980) dB dλ − dA dz = [A, B] A = A(λ; z, w, wz), B = B(λ; z, w, wz) ∈ sl2. This means that the monodromy data of the linear system dY dλ = A(λ; z, w, wz)Y are locally constant along solutions of the Painlev´ e equation. The monodromy data play the role of initial conditions.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

All Painlev´ e equations are isomonodromic deformation equations (Jimbo-Miwa1980) dB dλ − dA dz = [A, B] A = A(λ; z, w, wz), B = B(λ; z, w, wz) ∈ sl2. This means that the monodromy data of the linear system dY dλ = A(λ; z, w, wz)Y are locally constant along solutions of the Painlev´ e equation. The monodromy data play the role of initial conditions. The monodromy data are encoded in an affine cubic surfaces called monodromy varieties.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Painlev´ e monodromy manifolds Saito and van der Put

Mφ := Spec(C[x1, x2, x3]/ < φ = 0 >) PVI x1x2x3 + x2

1 + x2 2 + x2 3 + ω1x1 + ω2x2 + ω3x3 = ω4

PV x1x2x3 + x2

1 + x2 2 + ω1x1 + ω2x2 + ω3x3 = ω4

PIV x1x2x3 + x2

1 + ω1x1 + ω2x2 + ω2x3 + 1 = ω4

PIII x1x2x3 + x2

1 + x2 2 + ω1x1 + ω2x2 = ω1 − 1

PII x1x2x3 + x1 + x2 + x3 = ω4 PI x1x2x3 + x1 + x2 + 1 = 0

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-I

Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P1 with simple poles. dY dλ = (A1(z) λ + A2(z) λ − t + A3(z) λ − 1 , ) Y , λ ∈ C \ {0, t, 1} (1) where A1, A2, A3 ∈ sl2(C), A1 + A2 + A3 = −A∞, diagonal.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-I

Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P1 with simple poles. dY dλ = (A1(z) λ + A2(z) λ − t + A3(z) λ − 1 , ) Y , λ ∈ C \ {0, t, 1} (1) where A1, A2, A3 ∈ sl2(C), A1 + A2 + A3 = −A∞, diagonal. Fundamental matrix: Y∞(λ) = (1 + O( 1

λ))λA∞.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-I

Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P1 with simple poles. dY dλ = (A1(z) λ + A2(z) λ − t + A3(z) λ − 1 , ) Y , λ ∈ C \ {0, t, 1} (1) where A1, A2, A3 ∈ sl2(C), A1 + A2 + A3 = −A∞, diagonal. Fundamental matrix: Y∞(λ) = (1 + O( 1

λ))λA∞.

Monodromy matrices γj(Y∞) = Y∞Mj

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-I

Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P1 with simple poles. dY dλ = (A1(z) λ + A2(z) λ − t + A3(z) λ − 1 , ) Y , λ ∈ C \ {0, t, 1} (1) where A1, A2, A3 ∈ sl2(C), A1 + A2 + A3 = −A∞, diagonal. Fundamental matrix: Y∞(λ) = (1 + O( 1

λ))λA∞.

Monodromy matrices γj(Y∞) = Y∞Mj Describes by generators of the fundamental group under the anti-isomorphism ρ : π1 ( P1\{0, t, 1, ∞}, λ1 ) → SL2(C).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-II

eigen(Mj) = eigen(exp(2πiAj) We fix the base point λ1 at infinity and the generators of the fundamental group to be γ1, γ2, γ3 such that γj encircles only the pole i once and are oriented in such a way that M1M2M3M∞ = I, M∞ = exp(2πiA∞). (2) Eigenvalues of Aj are (θj, −θj), j = 0, t, 1, ∞. α := (θ∞ − 1/2)2; β := −θ2

0;

γ := θ2

1;

δ := (1/4 − θt)2.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-III

Let: Gj := Tr(Mj) = 2 cos(πθj), j = 0, t, 1, ∞, The Riemann-Hilbert correspondence F(θ0, θt, θ1, θ∞)/G → M(G1, G2, G3, G∞)/SL2(C), where G is the gauge group, is defined by associating to each Fuchsian system its monodromy representation class. The representation space M(G1, G2, G3, G∞) is realised as an affine cubic surface (Jimbo) x1x2x3 + x2

1 + x2 2 + x2 3 + ω1x1 + ω2x2 + ω3x3 + ω4 = 0,

(3) where:

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI as isomonodromic deformation-IV

x1 = Tr (M2M3) , x2 = Tr (M1M3) , x3 = Tr (M1M2) . and −ωi := GkGj + GiG∞, i ̸= k, j, ω∞ = G 2

1 + G 2 2 + G 2 3 + G 2 ∞ + G1G2G3G∞ − 4.

Iwasaki proved that the triple (x1, x2, x3) satisfying the cubic relation (3) provides a set of coordinates on a large open subset S ⊂ M(G1, G2, G3, G∞). In what follows, we restrict to such open set.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

General Affine Cubic

The main object studied in this talk is the affine irreducible cubic surface Mφ := Spec(C[x1, x2, x3]/⟨φ=0⟩) where φ = x1x2x3+ϵ(d)

1 x2 1+ϵ(d) 2 x2 2+ϵ(d) 3 x2 3+ω(d) 1 x1+ω(d) 2 x2+ω(d) 3 x3+ω(d) 4

= 0, (4) According to Saito and Van der Put, all monodromy manifolds M(d) have the form of Mφ for φ form the list above.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Affine Cubic as it is -1:

In singularity theory - the universal unfolding of the D4 singularity.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Affine Cubic as it is -1:

In singularity theory - the universal unfolding of the D4 singularity. Oblomkov: the quantisation of the D4 affine cubic surface Mφ coincides with spherical subalgebra of the generalised rank 1 double affine Hecke algebra H (or Cherednick algebra of type C1C ν

1 )

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Affine Cubic as it is -1:

In singularity theory - the universal unfolding of the D4 singularity. Oblomkov: the quantisation of the D4 affine cubic surface Mφ coincides with spherical subalgebra of the generalised rank 1 double affine Hecke algebra H (or Cherednick algebra of type C1C ν

1 )

In algebraic geometry - projective completion: M

φ := {(u, v, w, t) ∈ P3 |x2 1t + x2 2t + x2 3t − x1x2x3+

+ω3x1t2 + ω2x2t2 + ω3x3t2 + ω4t3 = 0} is a del Pezzo surface of degree three and differs from it by three smooth lines at infinity forming a triangle [Oblomkov] t = 0, x1x2x3 = 0.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

This family of cubics is a variety Mφ = {(¯ x, ¯ ω) ∈ C3 × Ω) : φ(¯ x, ¯ ω) = 0} where ¯ x = (x1, x2, x3), ¯ ω = (ω1, ω2, ω3, ω4) and the ”¯ x−forgetful” projection π : Mφ → Ω : π(¯ x, ¯ ω) = ¯ ω. This projection defines a family of affine cubics with generically non–singular fibres π−1(¯ ω) The cubic surface Mφ has a volume form ϑ¯

ω given by the Poincar´

e residue formulae: ϑ¯

ω =

dx1 ∧ dx2 (∂φ¯

ω)/(∂x3) =

dx2 ∧ dx3 (∂φ¯

ω)/(∂x1) =

dx3 ∧ dx1 (∂φ¯

ω)/(∂x2).

(5)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The volume form is a holomorphic 2-form on the non-singular part

• f Mφ and it has singularities along the singular locus. This form

defines the Poisson brackets on the surface in the usual way as {x1, x2}¯

ω = ∂φ¯ ω

∂x3 (6) The other brackets are defined by circular transposition of x1, x2, x3. For (i, j, k) = (1, 2, 3): {xi, xj}¯

ω = ∂φ¯ ω

∂xk = xixj + 2ϵd

i xk + ωd i

(7) and the volume form (5) reads as ϑ¯

ω =

dxi ∧ dxj (∂φ¯

ω/∂xk) =

dxi ∧ dxj (xixj + 2ϵd

i xk + ωd i ).

(8)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Observe that for any φ ∈ C[x1, x2, x3] the following formulae define a Poisson bracket on C[x1, x2, x3]: {xi, xi+1} = ∂φ ∂xi+2 , xi+3 = xi, (9) and φ itself is a central element for this bracket, so that the quotient space Mφ := Spec(C[x1, x2, x3]/⟨φ=0⟩) inherits the Poisson algebra structure [Nambu ∼ 70]. Today we are going to re-parametrize and to quantize it.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Affine Cubic as it is -2:

In the Painlev´ e context (the dynamic on the family of PVI monodromy surfaces) this cubic was considered by S. Cantat et F. Loray and by M. Inaba, K. Iwasaki and M.Saito.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Affine Cubic as it is -2:

In the Painlev´ e context (the dynamic on the family of PVI monodromy surfaces) this cubic was considered by S. Cantat et F. Loray and by M. Inaba, K. Iwasaki and M.Saito. PVI ( ˜ D4) cubic with only ω4 ̸= 0 admits the log-canonical symplectic structure ¯ ϑ = du∧dv

uv

under isomorphism C∗ × C∗/ı → Mφ by (u, v) → (x1 = −(u + 1 u ), x2 = −(v + 1 v ), x3 = −(uv + 1 uv )) and ı : C∗ → C∗ is the involution ı(u) = 1

u, ı(v) = 1 v .

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Affine Cubic as it is -2:

In the Painlev´ e context (the dynamic on the family of PVI monodromy surfaces) this cubic was considered by S. Cantat et F. Loray and by M. Inaba, K. Iwasaki and M.Saito. PVI ( ˜ D4) cubic with only ω4 ̸= 0 admits the log-canonical symplectic structure ¯ ϑ = du∧dv

uv

under isomorphism C∗ × C∗/ı → Mφ by (u, v) → (x1 = −(u + 1 u ), x2 = −(v + 1 v ), x3 = −(uv + 1 uv )) and ı : C∗ → C∗ is the involution ı(u) = 1

u, ı(v) = 1 v .

The family (3) can be ”uniformize” by some analogues of theta-functions related to toric mirror data on log-Calabi-Yau surfaces (M. Gross, P. Hacking and S.Keel (see Example 5.12 of ”Mirror symmetry for log-Calabi-Yau varieties I, arXiv:1106.4977).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The other Painlev´ e equations

The PVI monodromy manifold is the SL2(C)–character variety

• f a four holed Riemann sphere.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The other Painlev´ e equations

The PVI monodromy manifold is the SL2(C)–character variety

• f a four holed Riemann sphere.

What are the underlying Riemann surfaces for the other Painlev´ e equations?

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The other Painlev´ e equations

The PVI monodromy manifold is the SL2(C)–character variety

• f a four holed Riemann sphere.

What are the underlying Riemann surfaces for the other Painlev´ e equations? Is there a ”toric” (or cluster algebra) structure on it?

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The other Painlev´ e equations

The PVI monodromy manifold is the SL2(C)–character variety

• f a four holed Riemann sphere.

What are the underlying Riemann surfaces for the other Painlev´ e equations? Is there a ”toric” (or cluster algebra) structure on it? Use the confluence scheme of the Painlev´ e equations.

P D6

III

• P D7

III

• P D8

III

PV I PV

• P deg

V

• P JM

II

PI PIV

• P F N

II

• Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´

e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

”Confluented ”Poisson algebras

PoissD6

III

• PoissD7

III

• PoissD8

III

• PoissVI
• PoissV
• Poissdeg

V

• PoissJM

II

PoissI

• PoissIV
• PoissFN

II

• Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´

e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Poincar´ e-Katz invariants and Stokes rays

Painlev´ e eqs

• no. of cusps

Katz invariants

• no. Stokes rays

pole–orders for φ PVI (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0) (2, 2, 2, 2) PV (0, 0, 2) (0, 0, 1) (0, 0, 2) (2, 2, 4) PVdeg (0, 0, 1) (0, 0, 1/2) (0, 0, 1) (2, 2, 3) PIV (0, 4) (0, 2) (0, 4) (2, 6) PIII D6 (0, 2, 2) (0, 1, 1) (0, 2, 2) (2, 4, 4) PIII D7 (0, 1, 2) (0, 1/2, 1) (0, 1, 2) (2, 3, 4) PIII D8 (0, 1, 1) (0, 1/2, 1/2) (0, 1, 1) (2, 3, 3) PII FN (0, 3) (0, 3/2) (0, 3) (2, 5) PII MJ 6 3 6 8 PI 5 5/2 5 7

Table: For each Painlev´ e isomonodromic problem, this table displays the number of cusps on each hole for the corresponding Riemann surface, the Katz invariants associated to the corresponding flat connection, the number of Stokes rays in the linear system defined by the flat connection and the number of poles of the quadratic differential φ defined by the linear system.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Diagonalizable (non-ramified) case

Notation: the fundamental matrix at an irregular singular point λk has the form Yk = Gk(λ)λΛk (eQk(λ) e−Qk(λ) )

A(λ) Casimirs extended exponents dim(C) PV

A0 λ + A1 λ−1 + A∞

eigen(A0),eigen(A1),Λ∞ Q∞ = t

2λ

7 PIV

A0 λ + A1 + A∞λ

eigen(A0),Λ∞ Q∞ = λ2 + t

2λ

8 PIII D6

A0 λ2 + A1 λ + A∞

Λ0, Λ∞ Q∞ = t

2λ,Q0 = t 2 1 λ

8 PII MJ A0 + A1λ + A∞λ2 Λ∞ Q∞ = λ3 + t

2λ

9

Table: Here Qk(λ) is polynomial in (λ − λk) of order n − 1 with n being the order of λk and Λk is the formal monodromy (diagonal). Expand A(λ) near λk to calculate Qk(λ) and Λk, then diagonalize it using the gauge transformation Gk(λ)..

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Non-diagonalizable (ramified) case

The same notations.

A(λ) Casimirs extended exponents dim(C) PVdeg

A0 λ + A1 λ−1 + A∞

eigen(A0),eigen(A1),Λ∞ Q∞ = t

2

√ λ 5 PIII D7

A0 λ2 + A1 λ + A∞

Λ0, Λ∞ Q0 =

1 √ λ, Q∞ = t 2λ

6 PIII D8

A0 λ2 + A1 λ + A∞

Λ0, Λ∞ Q∞ = √ λ, Q0 =

1 √ λ

4 PII FN

A0 λ + A1 + A∞λ

eigen(A0), Λ∞ Q∞ = λ3/2 + t

2

√ λ 6 PI A0 + A1λ + A∞λ2 Λ∞ Q∞ = λ5/2 + t

2

√ λ 7

Table: Here dim PVdeg = dim PV − 2, dim PIII D7 = dim PIII − 2, dim PIII D8 = dim PIII D7 − 2, dim PII FN = dim PIV − 2, dim PI = dim PII JM − 2.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Geometric confluence scheme of the Painlev´ e equations:

• application. (P. Gavrylenko, O. Lisovyy, arXiv:1608.00958)

VI V Vdeg III(D ) III(D ) III(D ) IV IIFN IIJM I

6 7 8

Figure 3: CMR confluence diagram for Painlevé equations. Whittaker Bessel Gauss Figure 4: Some solvable RHPs in rank N = 2: Gauss hypergeometric (3 regular punc- tures), Whittaker (1 regular + 1 of Poincaré rank 1) and Bessel (1 regular + 1 of rank 1

2).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas:history

The character variety of a Riemann sphere with 4 holes Hom(π1(P1 \ {0, t, 1, ∞}); SL2(C))/SL2(C) is the monodromy cubic of the Painlev´ e VI (Goldman-Toledo).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas:history

The character variety of a Riemann sphere with 4 holes Hom(π1(P1 \ {0, t, 1, ∞}); SL2(C))/SL2(C) is the monodromy cubic of the Painlev´ e VI (Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections (Ph. Boalch 2011-2015).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas:history

The character variety of a Riemann sphere with 4 holes Hom(π1(P1 \ {0, t, 1, ∞}); SL2(C))/SL2(C) is the monodromy cubic of the Painlev´ e VI (Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections (Ph. Boalch 2011-2015). Quasi-Poisson structures (A. Alexeev, Y. Kosmann-Schwarzbach, E.Meinrenken, M. Van den Bergh 1999 -2005).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas:history

The character variety of a Riemann sphere with 4 holes Hom(π1(P1 \ {0, t, 1, ∞}); SL2(C))/SL2(C) is the monodromy cubic of the Painlev´ e VI (Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections (Ph. Boalch 2011-2015). Quasi-Poisson structures (A. Alexeev, Y. Kosmann-Schwarzbach, E.Meinrenken, M. Van den Bergh 1999 -2005). Moduli spaces for quilted surfaces. (D. Li-Bland, P. Severa 2013).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas:history

The character variety of a Riemann sphere with 4 holes Hom(π1(P1 \ {0, t, 1, ∞}); SL2(C))/SL2(C) is the monodromy cubic of the Painlev´ e VI (Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections (Ph. Boalch 2011-2015). Quasi-Poisson structures (A. Alexeev, Y. Kosmann-Schwarzbach, E.Meinrenken, M. Van den Bergh 1999 -2005). Moduli spaces for quilted surfaces. (D. Li-Bland, P. Severa 2013). Stokes groupoides (M. Gualtieri, S. Li, B. Pym 2014).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas: Painlev´ e context

Poisson structures and Isomonodromic deformations on the sphere and torus (Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas: Painlev´ e context

Poisson structures and Isomonodromic deformations on the sphere and torus (Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV (J.P. Ramis, E. Paul. 2015).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas: Painlev´ e context

Poisson structures and Isomonodromic deformations on the sphere and torus (Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV (J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” (Chekhov-Mazzocco -R.2015).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas: Painlev´ e context

Poisson structures and Isomonodromic deformations on the sphere and torus (Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV (J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” (Chekhov-Mazzocco -R.2015). The real slice of the SL2(C) character variety is the Teichm¨ uller space.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas: Painlev´ e context

Poisson structures and Isomonodromic deformations on the sphere and torus (Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV (J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” (Chekhov-Mazzocco -R.2015). The real slice of the SL2(C) character variety is the Teichm¨ uller space. The shear coordinates on the Teichm¨ uller space can be complexified ⇒ coordinate description for the character variety.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Basic ideas: Painlev´ e context

Poisson structures and Isomonodromic deformations on the sphere and torus (Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV (J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” (Chekhov-Mazzocco -R.2015). The real slice of the SL2(C) character variety is the Teichm¨ uller space. The shear coordinates on the Teichm¨ uller space can be complexified ⇒ coordinate description for the character variety. To visualize the confluence and the ”decoration” we shall introduce two moves correspond to certain asymptotics in the (complexified) shear coordinates.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Two mouves

Hooking holes: Pinching two sides of the same hole:

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Figure: The process of confluence of two holes on the Riemann sphere with four holes. Chewing-gum move: hook two holes together and stretch to infinity by keeping the area between them finite (see Fig.). As a result we obtain a Riemann sphere with one less hole, but with two new cusps on the boundary of this hole. The red geodesic line which was initially closed becomes infinite, therefore two horocycles (the green dashed circles) must be introduced in order to measure its length.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Teichm¨ uller space

For Riemann surfaces with holes: Hom (π1(Σ), PSL2(R)) /GL2(R).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Teichm¨ uller space

For Riemann surfaces with holes: Hom (π1(Σ), PSL2(R)) /GL2(R). Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Teichm¨ uller space

For Riemann surfaces with holes: Hom (π1(Σ), PSL2(R)) /GL2(R). Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Teichm¨ uller space

For Riemann surfaces with holes: Hom (π1(Σ), PSL2(R)) /GL2(R). Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Develop bordered cusped Teichm¨ uller theory asymptotically.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Teichm¨ uller space

For Riemann surfaces with holes: Hom (π1(Σ), PSL2(R)) /GL2(R). Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Develop bordered cusped Teichm¨ uller theory asymptotically. This will provide a cluster algebra of geometric type

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Coordinates: geodesic lengths

Theorem The geodesic length functions (Gγ := Trγ = 2cosh(lγ)) form an algebra with multiplication: GγG˜

γ = Gγ˜ γ + Gγ˜ γ−1.

(Tr(AB) + Tr(AB−1) = Tr(A)Tr(B) ∀A, B ∈ SL2(C))

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Coordinates: geodesic lengths

Theorem The geodesic length functions (Gγ := Trγ = 2cosh(lγ)) form an algebra with multiplication: GγG˜

γ = Gγ˜ γ + Gγ˜ γ−1.

(Tr(AB) + Tr(AB−1) = Tr(A)Tr(B) ∀A, B ∈ SL2(C)) ˜ γ γ

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Coordinates: geodesic lengths

Theorem The geodesic length functions (Gγ := Trγ = 2cosh(lγ)) form an algebra with multiplication: GγG˜

γ = Gγ˜ γ + Gγ˜ γ−1.

(Tr(AB) + Tr(AB−1) = Tr(A)Tr(B) ∀A, B ∈ SL2(C)) ˜ γ γ =

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Coordinates: geodesic lengths

Theorem The geodesic length functions (Gγ := Trγ = 2cosh(lγ)) form an algebra with multiplication: GγG˜

γ = Gγ˜ γ + Gγ˜ γ−1.

(Tr(AB) + Tr(AB−1) = Tr(A)Tr(B) ∀A, B ∈ SL2(C)) ˜ γ γ = γ˜ γ−1 + γ˜ γ

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Poisson structure

{Gγ, G˜

γ} = 1

2Gγ˜

γ − 1

2Gγ˜

γ−1.

{ ˜ γ γ } = 1

2

γ−1˜ γ − 1

2

γ˜ γ

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation-1

˜ γa ˜ γc ˜ γb ˜ γd ˜ γe ˜ γf G˜

γeG˜ γf = G˜ γaG˜ γc + G˜ γbG˜ γd

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation-2

aa′ + bb′ = cc′ c a b c′ a′ b′

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x1 x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

x1x′

1 = x9x7 + x8x2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x1 x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

x1x′

1 = x9x7 + x8x2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x1 x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

x1x′

1 = x9x7 + x8x2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x1 x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

x1x′

1 = x9x7 + x8x2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

x1x′

1 = x9x7 + x8x2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

(x′

1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′ 1, x2, x′ 3, x4, x5, x6, x7, x8, x9)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Ptolemy Relation

• x′

1

x2 x′

2

x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

(x′

1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′ 1, x′ 2, x3, x4, x5, x6, x7, x8, x9)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Cluster algebra

We call a set of n numbers (x1, . . . , xn) a cluster of rank n.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Cluster algebra

We call a set of n numbers (x1, . . . , xn) a cluster of rank n. A seed consists of a cluster and an exchange matrix B, i.e. a skew–symmetrisable matrix with integer entries.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Cluster algebra

We call a set of n numbers (x1, . . . , xn) a cluster of rank n. A seed consists of a cluster and an exchange matrix B, i.e. a skew–symmetrisable matrix with integer entries. A mutation is a transformation µi : (x1, x2, . . . , xn) → (x′

1, x′ 2, . . . , x′ n), µi : B → B′ where

xix′

i =

∏

j:bij>0

xbij

j

+ ∏

j:bij<0

x−bij

j

, x′

j = xj ∀j ̸= i.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Cluster algebra

We call a set of n numbers (x1, . . . , xn) a cluster of rank n. A seed consists of a cluster and an exchange matrix B, i.e. a skew–symmetrisable matrix with integer entries. A mutation is a transformation µi : (x1, x2, . . . , xn) → (x′

1, x′ 2, . . . , x′ n), µi : B → B′ where

xix′

i =

∏

j:bij>0

xbij

j

+ ∏

j:bij<0

x−bij

j

, x′

j = xj ∀j ̸= i.

Definition A cluster algebra of rank n is a set of all seeds (x1, . . . , xn, B) related to one another by sequences of mutations µ1, . . . , µk. The cluster variables x1, . . . , xk are called exchangeable, while xk+1, . . . , xn are called frozen. [Fomin-Zelevnsky 2002].

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Example

Cluster algebra of rank 9 with 3 exchangeable variables x1, x2, x3 and 6 frozen ones x4, . . . , x9.

• x′

1

x1 x2 x3 x4 x5 x6 x7 x8 x9 (x1, x2, x3, x4, x5, x6, x7, x8, x9) → (x′

1, x2, x3, x4, x5, x6, x7, x8, x9)

x1x′

1 = x9x7 + x8x2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Outline

Are all cluster algebras of geometric origin?

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Outline

Are all cluster algebras of geometric origin? Introduce bordered cusps

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Outline

Are all cluster algebras of geometric origin? Introduce bordered cusps Geodesics length functions on a Riemann surface with bordered cusps form a cluster algebra.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Outline

Are all cluster algebras of geometric origin? Introduce bordered cusps Geodesics length functions on a Riemann surface with bordered cusps form a cluster algebra. All Berenstein-Zelevinsky cluster algebras are geometric

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Poisson bracket

Introduce cusped laminations Compute Poisson brackets between arcs in the cusped lamination.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Poisson bracket

Introduce cusped laminations Compute Poisson brackets between arcs in the cusped lamination.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Poisson structure

Theorem The Poisson algebra of the λ-lengths of a complete cusped lamination is a Poisson cluster algebra [Chekhov-Mazzocco. ArXiv:1509.07044]. {gsi,tj, gpr,ql} = gsi,tjgpr,qlIsi,tj,pr,ql Isi,tj,pr,ql = ϵi−rδs,p+ϵj−rδt,p+ϵi−lδs,q+ϵj−lδt,q

4

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-1

What is the character variety of a Riemann surface with cusps on its boundary?

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-1

What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom (π1(Σ), PSL2(C)) /GL2(C).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-1

What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom (π1(Σ), PSL2(C)) /GL2(C). For Riemann surfaces with bordered cusps: Decorated character variety [Chekhov-Mazzocco-V.R. arXiv:1511.03851]

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-1

What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom (π1(Σ), PSL2(C)) /GL2(C). For Riemann surfaces with bordered cusps: Decorated character variety [Chekhov-Mazzocco-V.R. arXiv:1511.03851] Replace π1(Σ) with the groupoid G of all paths γij from cusp i to cusp j modulo homotopy.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-1

What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom (π1(Σ), PSL2(C)) /GL2(C). For Riemann surfaces with bordered cusps: Decorated character variety [Chekhov-Mazzocco-V.R. arXiv:1511.03851] Replace π1(Σ) with the groupoid G of all paths γij from cusp i to cusp j modulo homotopy. Replace tr by two characters: tr and trK.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Cusped fat graphs

Cusped fat graph (a graph with the prescribed cyclic ordering of edges entering each vertex) Gg,s,n a spine of the Riemann surface Σg,s,n with g handles, s holes and n > 0 decorated bordered cusps if (a) this graph can be embedded without self-intersections in Σg,s,n; (b) all vertices of Gg,s,n are three-valent except exactly n

• ne-valent vertices (endpoints of the open edges), which are

placed at the corresponding bordered cusps; (c) upon cutting along all non-open edges of Gg,s,n the Riemann surface Σg,s,n splits into s polygons each containing exactly

• ne hole and being simply connected upon contracting this

hole.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Geometric laminations

We call geometric cusped geodesic lamination (CGL) on a bordered cusped Riemann surface a set of nondirected curves up to a homotopy equivalence such that (a) these curves are either closed curves (γ) or arcs (a) that start and terminate at bordered cusps (which can be the same cusp); (b) these curves have no (self)intersections inside the Riemann surface (but can be incident to the same bordered cusp); (c) these curves are not empty loops or empty loops starting and terminating at the same cusp.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-2

We introduce the fundamental groupoid of arcs G as the set of all directed paths γij : [0, 1] → ˜ Σg,s,n such that γij(0) = mi and γij(1) = mj modulo homotopy. The groupoid structure is dictated by the usual path–composition rules. For each mj, j = 1, . . . , n, the isotopy group Πj = {γjj|γjj : [0, 1] → ˜ Σg,s,n, γjj(0) = mj, γjj(1) = mj}/{homotopy} is isomorphic to the usual fundamental group and Πj = γ−1

ij Πiγij

for any arc γij ∈ G. The decoration assigns to each arc γij a matrix Mij ∈ SL2(R), for example Mij = X(kj)LX(zn)R · · · LX(z1)RX(ki).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Decorated character variety-3

To associate a matrix in SL2(C) - complexify the coordinates. The decorated character variety is: Hom (G, SL2(C)) /∏n

j=1 Bj,

where Bj is the (unipotent) Borel subgroup in SL2(C) (one Borel subgroup for each cusp) with the characters: TrK : SL2(C) → C M → Tr(MK), where K = ( −1 ) .

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

PVI

s3 p3 s1 p1 s2 p2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x1; the solid geodesics are x2 and x3.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Shear coordinates in the Teichm¨ uller space

Fatgraph:

s3 p3 s1 p1 s2 p2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x1; the solid geodesics are x2 and x3.

Decompose each hyperbolic element in Right, Left and Edge matrices Fock, Thurston R := ( 1 1 −1 ) , L := ( 1 −1 −1 ) , Xy := ( − exp (y

2

) exp ( − y

2

) ) .

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

s3 p3 s1 p1 s2 p2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x1; the solid geodesics are x2 and x3.

The three geodesic lengths: xi = Tr(γjk) x1 = es2+s3 +e−s2−s3 +e−s2+s3 +(e

p2 2 +e− p2 2 )es3 +(e p3 2 +e− p3 2 )e−s2

x2 = es3+s1 +e−s3−s1 +e−s3+s1 +(e

p3 2 +e− p3 2 )es1 +(e p1 2 +e− p1 2 )e−s3

x3 = es1+s2 +e−s1−s2 +e−s1+s2 +(e

p1 2 +e− p1 2 )es2 +(e p2 2 +e− p2 2 )e−s1 Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

s3 p3 s1 p1 s2 p2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x1; the solid geodesics are x2 and x3.

The three geodesic lengths: xi = Tr(γjk) x1 = es2+s3 +e−s2−s3 +e−s2+s3 +(e

p2 2 +e− p2 2 )es3 +(e p3 2 +e− p3 2 )e−s2

x2 = es3+s1 +e−s3−s1 +e−s3+s1 +(e

p3 2 +e− p3 2 )es1 +(e p1 2 +e− p1 2 )e−s3

x3 = es1+s2 +e−s1−s2 +e−s1+s2 +(e

p1 2 +e− p1 2 )es2 +(e p2 2 +e− p2 2 )e−s1

{x1, x2} = x1x2 + 2x3 + ω3, {x2, x3} = x2x3 + 2x1 + ω1, {x3, x1} = x3x1 + 2x2 + ω2.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The confluence from the cubic associated to PVI to the one associated to PV is realized by p3 → p3 − 2 log[ϵ], in the limit ϵ → 0. We obtain the following shear coordinate description for the PV cubic: x1 = −es2+s3+ p2

2 + p3 2 − G3es2+ p2 2 ,

x2 = −es3+s1+ p3

2 + p1 2 − es3−s1+ p3 2 − p1 2 − G3e−s1− p1 2 − G1es3+ p3 2 ,

x3 = −es1+s2+ p1

2 + p2 2 − e−s1−s2− p1 2 − p2 2 − es1−s2+ p1 2 − p2 2 − G1e−s2− p2 2 − G

where Gi = e

pi 2 +e− pi 2 ,

i = 1, 2, G3 = e

p3 2 ,

G∞ = es1+s2+s3+ p1

2 + p2 2 + p3 2 . Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

These coordinates satisfy the following cubic relation: x1x2x3 + x2

1 + x2 2 − (G1G∞ + G2G3)x1 − (G2G∞ + G1G3)x2 −

−G3G∞x3 + G 2

∞ + G 2 3 + G1G2G3G∞ = 0.

(11) Note that the parameter p3 is now redundant, we can eliminate it by rescaling. To obtain the correct PV- cubic, we need to pick p3 = −p1 − p2 − 2s1 − 2s2 − 2s3 so that G∞ = 1.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

These coordinates satisfy the following cubic relation: x1x2x3 + x2

1 + x2 2 − (G1G∞ + G2G3)x1 − (G2G∞ + G1G3)x2 −

−G3G∞x3 + G 2

∞ + G 2 3 + G1G2G3G∞ = 0.

(11) Note that the parameter p3 is now redundant, we can eliminate it by rescaling. To obtain the correct PV- cubic, we need to pick p3 = −p1 − p2 − 2s1 − 2s2 − 2s3 so that G∞ = 1. {x1, x2} = x1x2 − G3G∞, {x2, x3} = x2x3 + 2x1 − (G1G∞ + G2G3), {x3, x1} = x3x1 + 2x2 − (G2G∞ + G1G3).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Geometrically speaking, sending the perimeter p3 to infinity means that we are performing a chewing-gum move: two holes, one of perimeter p3 and the other of perimeter s1 + s2 + s3 + p1

2 + p2 2 + p3 2 , become infinite, but the area between

them remains finite. This is leading to a Riemann sphere with three holes and two cusps

• n one of them. In terms of the fat-graph, this is represented by

Figure 2. The geodesic x3 corresponds to the closed loop obtained going around p1 and p2 (green and red loops), while x1 and x2 are ”asymptotic geodesics” starting at one cusp, going around p1 and p2 respectively, and coming back to the other cusp.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

s3 s1 p1 s2 p2 k2 k1 Figure 5. The fat graph corresponding to PV. The geodesic x3 remains closed, while x1 (the dashed line) and x2 become arcs.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

s3 s1 p1 s2 p2 k2 k1 Figure 5. The fat graph corresponding to PV. The geodesic x3 remains closed, while x1 (the dashed line) and x2 become arcs.

γb = X(k1)RX(s3)RX(s2)RX(p2)RX(s2)LX(s3)LX(k1)- BUT its length is b = trK(γb) = tr(bK), K = ( −1 )

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

s3 s1 p1 s2 p2 k2 k1 Figure 5. The fat graph corresponding to PV. The geodesic x3 remains closed, while x1 (the dashed line) and x2 become arcs.

{gsi,tj, gpr,ql} = gsi,tjgpr,ql

ϵi−rδs,p+ϵj−rδt,p+ϵi−lδs,q+ϵj−lδt,q 4

{b, d} = {g13,14, g21,18} = g13,14g21,18 ϵ3−1δ1,2 + ϵ4−1δ1,2 + ϵ3−8δ1,1 + ϵ4−8δ1,1 4 = −bd 1 2

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The character variety of a Riemann sphere with three holes and two cusps on one boundary is 7-dimensional (rather than 2-dimensional like in PVI case). The fat-graph admits a complete cusped lamination as displayed in the figure below. A full set of coordinates on the character variety is given by the five elements in the lamination and the two parameters G1 and G2 which determine the perimeter of the two non-cusped holes.

d c a b e k1 k2 s3 s1 s2 p2 p1 Figure 6. The system of arcs for PV. Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Notice that there are two shear coordinates associated to the two cusps, they are denoted by k1 and k2, their sum corresponds to what we call p3 above. These shear coordinates do not commute with the other ones, they satisfy the following relations: {s3, k1} = {k1, k2} = {k2, s3} = 1. As a consequence in the character variety, the elements G3 and G∞ are not Casimirs. In terms of shear coordinates, the elements in the lamination are expressed as follows: a = ek1+s1+2s2+s3+ p1

2 +p2,

b = ek1+s2+s3+ p2

2 ,

e = e

k1 2 + k2 2 ,

c = ek1+s1+s2+s3+ p1

2 + p2 2 ,

d = e

k1 2 + k2 2 +s1+s2+s3+ p1 2 + p2 2 .

(12)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

They satisfy the following Poisson relations: a {a, b} = ab, {a, c} = 0, {a, d} = −1 2ad, {a, e} = 1 2ae, (13) {b, c} = 0, {b, d} = −1 2bd, {b, e} = 1 2be, (14) {c, d} = −1 2cd, {c, e} = 1 2ce, {d, e} = 0, (15) so that the element G3G∞ = de is a Casimir.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The symplectic leaves are determined by the values of the three Casimirs G1, G2 and G3G∞. On each symplectic leaf, the PV monodromy manifold (11) is the subspace defined by those functions of a, b, c (and of the Casimir values G1, G2, G3G∞) which commute with G3 = e. To see this, we can use relations (12) to determine the exponentiated shear coordinates in terms of a, b, c, d, e and then deduce he expressions

• f x1, x2, x3 in terms of the lamination. We obtain the following

expressions: x1 = −e a c − d b c , x2 = −e b c − G1d b a − d b2 ac − d c a, (16) x3 = −G2 c b − G1 c a − b a − c2 ab − a b, (17) which automatically satisfy (11).

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Due to the Poisson relations (13) the functions that commute with e are exactly the functions of a

b, b c , c

• a. Such functions may depend
• n the Casimir values G1, G2 and G3G∞ and e itself, so that

d = G∞ becomes a parameter now. With this in mind, it is easy to prove that x1, x2, x3 are algebraically independent functions of

a b, b c , c a so that x1, x2, x3 form a basis in the space of functions

which commute with e.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Due to the Poisson relations (13) the functions that commute with e are exactly the functions of a

b, b c , c

• a. Such functions may depend
• n the Casimir values G1, G2 and G3G∞ and e itself, so that

d = G∞ becomes a parameter now. With this in mind, it is easy to prove that x1, x2, x3 are algebraically independent functions of

a b, b c , c a so that x1, x2, x3 form a basis in the space of functions

which commute with e. It is worth reminding that the exponentials

• f the shear coordinates satisfy the log-canonical Poisson bracket.

The ”reduced” 2D decorated character variety is the affine cubic family: π : Spec(C[G1, G2, G3, G−1

3 , x1, x2, x3]/x1x2x3 + x2 1 + x2 2 −

−(G1 + G2G3)x1 − (G2 + G1G3)x2 − G3x3 + 1 + G 2

3 + G1G2G3) −

→ Spec(C[G1, G2, G3, G−1

3 ]).

(18)

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

The decorated character variety associated with PII JM has 6 cusps

• n the boundary is 9-dimensional. The fat-graph admits a

complete cusped lamination as displayed in the figure below.

i a c d e f b g h k1 k2 k3 k4 s3 s2 s1 k5 k6 Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Quantisation

For standard geodesic lengths Gγ → G ℏ

γ

[Chekhov-Fock ’99]:

[ G ℏ

˜ γ

G ℏ

γ

] = q− 1

2

G ℏ

γ−1˜ γ

+ q

1 2

G ℏ

γ˜ γ

[G ℏ

γ , G ℏ ˜ γ ] = q− 1

2 G ℏ

γ−1˜ γ + q

1 2 G ℏ

γ˜ γ

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Quantisation

For standard geodesic lengths Gγ → G ℏ

γ

[Chekhov-Fock ’99]:

[ G ℏ

˜ γ

G ℏ

γ

] = q− 1

2

G ℏ

γ−1˜ γ

+ q

1 2

G ℏ

γ˜ γ

[G ℏ

γ , G ℏ ˜ γ ] = q− 1

2 G ℏ

γ−1˜ γ + q

1 2 G ℏ

γ˜ γ

For arcs gsi,tj → gℏ

si,tj:

qIsi ,tj ,pr ,ql gℏ

si,tjgℏ pr,ql = gℏ pr,qlgℏ si,tjqIpr ,ql ,si ,tj

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Quantisation

For standard geodesic lengths Gγ → G ℏ

γ

[Chekhov-Fock ’99]:

[ G ℏ

˜ γ

G ℏ

γ

] = q− 1

2

G ℏ

γ−1˜ γ

+ q

1 2

G ℏ

γ˜ γ

[G ℏ

γ , G ℏ ˜ γ ] = q− 1

2 G ℏ

γ−1˜ γ + q

1 2 G ℏ

γ˜ γ

For arcs gsi,tj → gℏ

si,tj:

qIsi ,tj ,pr ,ql gℏ

si,tjgℏ pr,ql = gℏ pr,qlgℏ si,tjqIpr ,ql ,si ,tj

This identifies the geometric basis of the quantum cluster algebras introduced by Berenstein - Zelevinsky.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Conclusion

New notion of decorated character variety

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Conclusion

New notion of decorated character variety In the case of the Painlev´ e differential equations, each decorated character variety is a Poisson manifold of dimension 3s + 2n − 6, where s is the number of holes and n ≥ 1 is the number of cusps.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Conclusion

New notion of decorated character variety In the case of the Painlev´ e differential equations, each decorated character variety is a Poisson manifold of dimension 3s + 2n − 6, where s is the number of holes and n ≥ 1 is the number of cusps. In each case the decorated character variety admits a special Poisson submanifold defined by the set of functions which Poisson commute with the frozen cluster variables.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Conclusion

New notion of decorated character variety In the case of the Painlev´ e differential equations, each decorated character variety is a Poisson manifold of dimension 3s + 2n − 6, where s is the number of holes and n ≥ 1 is the number of cusps. In each case the decorated character variety admits a special Poisson submanifold defined by the set of functions which Poisson commute with the frozen cluster variables. This submanifold is defined as a cubic surface

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Conclusion

New notion of decorated character variety In the case of the Painlev´ e differential equations, each decorated character variety is a Poisson manifold of dimension 3s + 2n − 6, where s is the number of holes and n ≥ 1 is the number of cusps. In each case the decorated character variety admits a special Poisson submanifold defined by the set of functions which Poisson commute with the frozen cluster variables. This submanifold is defined as a cubic surface By quantisation: quantum cluster algebra of geometric type.

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation

Painlev´ e equations Geodesic lengths Decorated character variety Quantisation

Conclusion

New notion of decorated character variety In the case of the Painlev´ e differential equations, each decorated character variety is a Poisson manifold of dimension 3s + 2n − 6, where s is the number of holes and n ≥ 1 is the number of cusps. In each case the decorated character variety admits a special Poisson submanifold defined by the set of functions which Poisson commute with the frozen cluster variables. This submanifold is defined as a cubic surface By quantisation: quantum cluster algebra of geometric type. Many thanks for your attention!!!

Volodya Roubtsov, ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ e d’Angers. Based on Chekhov-Mazzocco-R. Painlev´ e monodromy varieties: geometry and quantisation